Q2 of 15 Page 82

The picture on the right shows two circles centred on the same point and a line intersecting them.


Prove that the parts of the line between the circles on either side are equal.


D is the centre of both the circles.


DG is a perpendicular drawn on EF(HI).


We have to prove that, EH = IF.


In ΔDHG and ΔDIG we have,


DH = DI [radius of the same circle]


DG is the common side.


DG is perpendicular on HI.


∴ ∠DHG = ∠DIG = 90°


∴ ∆DHG≅∆DIG [SAS congruency]


∴ GH = GI …………… (1)


In ΔDEG and ΔDFG we have,


DE = DF [radius of the same circle]


DG is the common side.


DG is perpendicular on EF.


∴ ∠DEG = ∠DFG = 90°


∴ ∆DEG≅∆DFG [SAS congruency]


∴ GE = GF …………… (2)


From, (2) – (1) we get,


⇒ GE – GH = GF – GI


⇒ EH = IF


More from this chapter

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1

Prove that the line joining the centres of two intersecting circles is the perpendicular bisector of the line joining the points of intersection.

 3

A chord and the diameter through one of its ends are drawn in a circle. A chord of the same inclination is drawn on the other side of the diameter.


Prove that the chords are of the same length.

 4

Prove that the angle made by two equal chords drawn from a point on the circle is bisected by the diameter through that point.

 5

Draw a square and a circle through all four vertices. Draw diameters parallel to the sides of the square and draw a polygon joining the end points of these diameters and the vertices of the square.


Prove that this polygon is a regular octagon.